By virtue of Bourgain's Theorem C, where it is further assumed that \(|X| \le |A|\)  and the exponent of \(|X|\)  in the right hand side is \(1 -\delta \)  for some unspecified absolute positive \(\delta > 0\) , one could prove a weaker version of Theorem REF  where the hypothesis would be \(|A| = \Omega (p^{2/3-c})\)  for some absolute positive constant \(c>0\)  depending on unspecified absolute constants appearing in Bourgain's paper. Rudev's explicit version of Theorem C steams from a recent result of Aksoy-Yazici, Murphy, Rudnev, and Shkredov in [1]}, which is based on an adaption of a theorem of Guth and Katz from their solution to the Erdős distinct distance problem on the plane [2]} (see also [3]}, [4]}). It is analogous, albeit weaker, to a result of Murphy, Roche-Newton, and Shkredov over \(\mathbb {R}\)  [5]}.
